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I'm having trouble with this question:

Suppose there is a unique path between any pair of vertices in $G$. Prove that $G$ is a tree.

I know that a path is a trail where all vertices are distinct and a tree is a simple connected graph with no circuits.

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You need to define a tree since that statement is a definition of a tree. – Jacob Nov 13 '12 at 4:22
up vote 3 down vote accepted

HINT: Suppose that $G$ is not a tree; then there is a circuit in $G$. Pick two vertices, $u$ and $v$, in this circuit. Now use the circuit to find two different paths between $u$ and $v$.

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